6 Time preference

Saving, borrowing, investment and insurance decisions all involve intertemporal choice. Decisions are made at one point in time, with effects at another. The timing of the costs and benefits do not align.

People discount future costs and benefits. They prefer to receive benefits earlier, rather than later, and prefer to incur costs later rather than earlier.

We have already encountered discounting in Section 2.1, where an agent had the utility function:

\[U=u(C_0)+\beta u(C_1)\]

\[0 \leq \beta \leq 0\]

where $U$ is utility, and $C_0$ and $C_1$ are consumption in the first and second periods respectively. $\beta$ is a discount factor reflecting how much the agent weights consumption in the future relative to today.

A recap of some core concepts is below.

6.1 Exponential discounting

Exponential discounting occurs where future costs and benefits are discounted at a consistent rate through time. The following equation is an example of exponential discounting.

\[U=\sum_{t=0}^{t=T}\delta^t u(C_t)\]

\[0<\delta\leq 1\]

The degree of discounting in this equation evolves over time as 1, $\delta$, $\delta^2$, $\delta^3$, $\delta^4$ and so on. This results in a smooth decline in present value over time. Decisions made with exponential discounting are consistent over time.

6.2 Present bias

Present bias occurs when we place additional weight on costs and benefits at the present time. One simple model of present bias is the quasi-hyperbolic discounting model (it is a discrete time version of hyperbolic discounting).

\[U=u(C_0)+\sum_{t=1}^{t=T}\beta\delta^t u(C_t)\]

The degree of discounting in this equation evolves over time as 1, $\beta\delta$, $\beta\delta^2$, $\beta\delta^3$, $\beta\delta^4$ and so on. This progression results in a larger discount for the first period of delay ($\beta\delta$) than the degree of discount for each subsequent period of delay ($\delta$). There is a relative weighting toward the present.

Present bias of this nature can result in time inconsistency, with decisions at one point reversed at another if given the opportunity.

6.3 A numerical example

The following numerical example explores how an exponential discounter and a present-biased agent will each consider two choices:

Choice 1: Would this agent prefer $100 today ($t=0$) or $110 next week ($t=1$)?

Choice 2: Would this agent prefer $100 next week ($t=1$) or $110 in two weeks ($t=2$)?

I show that the exponential discounter will be consistent in their decisions through time, whereas the present-biased agent can be time inconsistent.

In this example I represent a stream of payoffs over time in the form $S=(t_1,x_1;t_2,x_2;...;t_n,x_n)$. For example, $(0,\$100)$ represents a payment of $100 at $t=0$, whereas $(0,\$100)$ represents a payment of $107 at $t=1$.

6.3.1 The exponential discounter

Suppose we have an exponential discounter with $\delta=0.95$ and utility each period of $u(x_n)=x_n$.

Choice 1: Would this agent prefer $100 today ($t=0$) or $110 next week ($t=1$)?

\[\begin{align*} U_0(0,\$100)&=u(\$100) \\[6pt] &=100 \\ \\ U_0(1,\$110)&=\delta u(\$110) \\[6pt] &=0.95*110 \\[6pt] &=104.50 \end{align*}\]

The exponential discounter will prefer to receive $110 next week.

Choice 2: Would this agent prefer $100 next week ($t=1$) or $110 in two weeks ($t=2$)?

\[\begin{align*} U_1(1,\$100)&=\delta u(\$110) \\ &=0.95*100 \\ &=95 \\ \\ U_1(2,\$110)&=\delta^2 u(\$110) \\ &=0.95^2*110 \\ &=99.275 \end{align*}\]

The exponential discounter will prefer to receive $110 in two weeks. The set of decisions across Choice 1 and Choice 2 are time consistent. If the agent selected $110 in two weeks for Choice 2 and was given a chance to change their choice after one week (which is effectively Choice 1), they would not change.

6.3.2 The present-biased agent

Suppose we have a present biased agent with $\delta=0.95$, $\beta=0.95$ and utility each period of $U(x_n)=x_n$.

Choice 1: Would this agent prefer $100 today ($t=0$) or $110 next week ($t=1$)?

\[\begin{align*} U_0(0,\$100)&=u(\$100)\\[6pt] &=100 \\ \\ U_0(1,\$110)&=\beta\delta u(\$110) \\[6pt] &=0.95*0.95*110 \\[6pt] &=99.275 \end{align*}\]

As $U_0(0,\$100) > U_0(1,\$110)$, the present-biased agent will prefer to receive $100 this week.

Choice 2: Would this agent prefer $100 next week ($t=1$) or $110 in two weeks ($t=2$)?

\[\begin{align*} U_0(1,\$100)&=\beta\delta u(\$100) \\[6pt] &=0.95*0.95*100 \\[6pt] &=90.25 \\ \\ U_0(2,\$110)&=\beta\delta^2 u(\$110) \\[6pt] &=0.95*0.95^2*110 \\[6pt] &=94.31 \end{align*}\]

As $U_0(1,\$100) < U_0(2,\$110)$, the present-biased agent will prefer to receive $110 in two weeks.

Putting those two choices together:

Choice 1: The present-biased agent will prefer $100 now to $110 in one week. Their preference for benefits now ($\beta$) leads them to prefer the immediate payoff.

Choice 2: The present-biased agent will prefer $110 in two weeks to $100 in one week. They are willing to wait longer for a larger reward, with both outcomes in the future and subject to the short-term discount ($\beta$).

Consider what would happen if they selected the $110 in two weeks in Choice 2, but after one week were asked if they would like to reconsider their choice. They are effectively being offered Choice 1. This would then lead them to change their mind and take the immediate $100.

This combination of decisions is time inconsistent. The agent’s actions are not consistent with their own initial plan.